Moment Closure for Local Control Models of Calcium-Induced Calcium Release in Cardiac Myocytes

¹ The College of William & Mary
² Department of Pharmacology and Systems Therapeutics
³ George Mason University

^* To whom correspondence should be addressed. E-mail: greg{at}as.wm.edu.

Submitted on November 19, 2007
Revised on December 23, 2007
Accepted on 18 April 2008

	Abstract

In prior work we introduced a probability density approach to modeling local control of Ca²⁺-induced Ca²⁺ release in cardiac myocytes [Williams et al., Biophys. J. 92(7):2311-28, 2007] where we derived coupled advection-reaction equations for the time-dependent bivariate probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum (SR) [Ca²⁺] conditioned on Ca²⁺ release unit (CaRU) state. When coupled to ODEs for the bulk myoplasmic and network SR [Ca²⁺], a realistic but minimal model of cardiac excitation-contraction coupling was produced that avoids the computationally demanding task of resolving spatial aspects of global Ca²⁺ signaling, while accurately representing heterogeneous local Ca²⁺ signals in a population of diadic subspaces and junctional SR depletion domains. Here we introduce a computationally effcient method for simulating such whole cell models when the dynamics of subspace [Ca²⁺] are much faster than those of junctional SR [Ca²⁺]. The method begins with the derivation of a system of ODEs describing the time-evolution of the moments of the univariate probability density functions for junctional SR [Ca²⁺] jointly distributed with CaRU state. This open system of ODEs is then closed using an algebraic relationship that expresses the third moment of junctional SR [Ca²⁺] in terms of the first and second moments. In simulated voltage-clamp protocols using 12-state CaRUs that respond to the dynamics of both subspace and junctional SR [Ca²⁺], this moment closure approach to simulating local control of excitation-contraction coupling produces high-gain Ca²⁺ release that is graded with changes in membrane potential, a phenomenon not exhibited by common pool models. Benchmark simulations indicate that the moment closure approach is nearly 10,000-times more computationally efficient than corresponding Monte Carlo simulations while leading to nearly identical results. We conclude by applying the moment closure approach to study the restitution of Ca²⁺-induced Ca²⁺ release during simulated two-pulse voltage-clamp protocols.

Key Words: calcium, cardiac, local control, moment closure, probability density